As far as Judaism is concerned Jesus was born to a Jewish mother and was Jewish. However, as you must be aware Judaism has no place for Jesus or his teachings Judaism categorically does not accept that Jesus could have been the Messiah or a prophet. There are a many reasons that Jews do not accept Jesus as the messiah, or mashiach, below are a few of them:
1. Genealogy: He was not descended from the House of David. According to Jewish law, tribal identification comes from the father's side, being Jewish, from the mother's side. According to Matthew 1, Joseph was descended from David (Although there are many contradictions between his genealogy there and that listed in Luke 3), however according to the same text, Joseph did not have sexual relations with Mary, therefore Jesus was not related to Joseph, and not a descendant of King David.
Three answers to this problem are given in classic Christian sources:
a. The genealogy is that of Mary - This is inadequate, since if he is claimed to be the Jewish messiah, and according to Jewish tradition he must be descended on his father's side, Mary's genealogy is irrelevant. b. He was adopted by Joseph -According to Jewish law, adoption does not change the status of the child. If an Israelite is adopted by a Cohen, (A descendant of Aaron the High Priest), the child does not become a Cohen, likewise if a descendant of David, adopts someone who is not, he does not become of the tribe of Judah and a descendant of David. c. It doesn't matter, he is a spiritual inheritor of King David - If it doesn't matter, why do Christian scriptures spend time establishing his genealogical pedigree? And if he is claimed to be the Jewish messiah, then according to Jewish tradition it does matter!
2. Messianic predictions: The main predictions concerning the Messiah are that he will bring peace to the world, gather the Jewish people from their exile to the land of Israel and rebuild the Temple in Jerusalem. After Jesus' appearance, the Temple was destroyed, the Jews were exiled all over the world and we have not even had one day of peace in the past 2,000 years. (Many of the wars in fact were started and fought by followers of Jesus) These events are enough to show that he was not the messiah.
The main Christian responses to these objections are: a. The Second Coming - First of all, we find this to be a contrived answer, since there is no mention of a second coming in the Jewish Bible. Second, why couldn't God accomplish His goals the first time round. Most importantly, the second coming idea is just an attempt at answering an obvious question but it certainly does not constitute proof of messianic claims. b. There is peace within his followers hearts - That is wonderful for them, but does that help the victims of the Inquisition, the Crusades, the Hundred Years War, the First World War, the Second World War etc. In each of the events that I mentioned most if not all the combatants, the violent oppressors and torturers where people who claimed to be followers of Jesus. And is peace in the heart a fulfillment of "swords into ploughshears etc."
3. Messiah's Qualifications: Messiah is a prophet, a scholar and a pious king. Jesus made a prediction that "The time is fulfilled, the kingdom of God is at hand." (Mark 1:15) That was 2000 years ago, has the kingdom of God come? Do you call the holocaust, Pol Pot and Stalin a world in which the kingdom of God has come? Jesus was not a great scholar - one of the requirements of the Messiah. (Travers Herford, The Pharisees, p. 206) Was Jesus a king? He was not anointed as king by a prophet (as was the rule in Jewish kings), he was not appointed by any judicial body as a leader and he did not rule over the Jewish people nor was he accepted by them. He was arrested, tortured and killed by the Romans like a common criminal. He had no army or government. The answer to my question i
